THE UNIVERSITY OF NEW SOUTH WALES
DEPARTMENT OF STATISTICS
Exercises for MATH3811, Statistical Inference
K Sample Problems. Measures of Association
1. The staff of a mental hospital is concerned with which kind of treatment is most effective for a
particular type of mental disorder. A battery of tests administered to all patients delineated a group
of 40 patients who were similar as regards diagnosis and also personality, intelligence, etc. These
people were randomly divided into four different groups of 10 each for treatment. These respective
groups received (1) electroshock, (2) psychoteraphy, (3) electroshock plus psychoteraphy, (4) no
treatment. At the end of this period the battery of tests was repeated on each patient. The only
type of measurement possible for these tests is a ranking of the 40 patients on the basis of their
relative degree of improvement at the end of the treatment period. Rank 1 indicates the highest level
of improvement , rank 2 the second highest etc. On the basis of these data (presented in the table
below), does there seem to be any difference in effectiveness of the types of treatment?
Group one Group two Group three Group four
19
14
12
38
22
21
1
39
25
2
5
40
24
6
8
30
29
10
4
31
26
16
13
32
37
17
9
33
23
11
15
36
27
18
3
34
28
7
20
35
Sum R:
260
122
90
348
2. The following data (measurements of skin potential (mV)) were reported when the emotions of fear,
happiness, depression, and calmness were requested from each of eight subjects (blocks). It is to
be tested that the average skin potential depends on which emotion is requested. The normality of
the data is seriously questioned. Carry out Friedman’s Test for randomized block experiment and
formulate your conclusion.
xij Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8
Fear
23.1
57.6
10.5
23.6
11.9
54.6
21
20.3
Happiness
22.7
53.2
9.7
19.6
13.8
47.1
13.6
23.6
Depression
22.5
53.7
10.8
21.1
13.7
39.2
13.7
16.3
Calmness
22.6
53.1
8.3
21.6
13.3
37.0
14.8
14.8
3. Twelve MBA graduates are studied to measure the strength of the relationship between their score
on the GMAT, which they took prior to entering graduate school, and their grade point average while
they were in the MBA program. Their GMAT scores and their GPAs are given below:
Student
1
2
3
4
5
6
7
8
9 10 11 12
GMAT 710 610 640 580 545 560 610 530 560 540 570 560
GPA
4
4 3.9 3.8 3.7 3.6 3.5 3.5 3.5 3.3 3.2 3.2
a) Compute Spearman’s ρ according to its original definition (this is the correct approach here since
there are ties) and according to the “shortcut formula” and compare the results. Test the null
hypothesis that the GPAs are independent of GMAT scores against the one-sided alternative
that high GPAs tend to be associated with high GMAT scores.
1
b) Calculate Kendall’s τ and perform the same test as in a) by utilizing the asymptotic distribution
of Kendall’s τ.
4. The data below gives the test scores of 13 nonidentical male twins. Test the hypothesis of independence versus the alternative that the twin’s test scores are positively correlated. Use Spearman’s ρ
and Kendall’s τ . Perform the calculations in S-Plus. Clearly formulate your conclusion.
Pair i
1
2
3
4
5
6
7
8
9 10 11 12 13
Twin Xi 277 169 157 139 108 213 232 229 114 232 161 149 128
Twin Yi 256 118 137 144 146 221 184 188 97 231 114 187 230
Answers
1. H = 31.89, df = 3, P-value < 0.001; reject H0 .
2. F = 6.45, df = 3, H0 is accepted at level α = 0.05
3. a) ρ = 0.59 (original formula), 0.5979 (shortcut formula). P-value: 0.025; reject H0 .
b) τ = 0.4355 (using the formula for the case of ties); p-value=0.032
4. cor.test(x1,x2,method=’spearman’,alternative=’greater’)
ρ = .5144, p-value=0.0382
cor.test(x1,x2,method=’kendall’,alternative=’greater’)
τ = .346, p-value=0.049
Decide there is (a marginal) evidence of a positive correlation in the scores of the nonidentical twins.
2